Modeling Material Stress Using Integrated Gaussian Markov Random Fields

TL;DR

This paper develops a sophisticated statistical model using Gaussian Markov random fields to analyze the spatial distribution of vonMises stress in tantalum grains, integrating physical geometry and advanced computational techniques for efficient Bayesian inference.

Contribution

It introduces a novel GMRF-based spatial model that incorporates grain boundary geometry and addresses computational challenges for large-scale Bayesian analysis.

Findings

Effective modeling of stress fields in complex grain geometries.

Successful implementation of scalable Bayesian inference techniques.

Robust handling of outliers in stress data.

Abstract

The equations of a physical constitutive model for material stress within tantalum grains were solved numerically using a tetrahedrally meshed volume. The resulting output included a scalar vonMises stress for each of the more than 94,000 tetrahedra within the finite element discretization. In this paper, we define an intricate statistical model for the spatial field of vonMises stress which uses the given grain geometry in a fundamental way. Our model relates the three-dimensional field to integrals of latent stochastic processes defined on the vertices of the one- and two-dimensional grain boundaries. An intuitive neighborhood structure of said boundary nodes suggested the use of a latent Gaussian Markov random field (GMRF). However, despite the potential for computational gains afforded by GMRFs, the integral nature of our model and the sheer number of data points pose substantial challenges for a full Bayesian analysis. To overcome these problems and encourage efficient exploration of the posterior distribution, a number of techniques are now combined: parallel computing, sparse matrix methods, and a modification of a block update strategy within the sampling routine. In addition, we use an auxiliary variables approach to accommodate the presence of outliers in the data.